Properties

Label 2-2200-1.1-c1-0-44
Degree $2$
Conductor $2200$
Sign $-1$
Analytic cond. $17.5670$
Root an. cond. $4.19131$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.339·3-s + 4.05·7-s − 2.88·9-s + 11-s − 4·13-s − 7.74·17-s − 7.06·19-s + 1.37·21-s + 2.72·23-s − 1.99·27-s − 4.73·29-s + 0.219·31-s + 0.339·33-s − 1.32·37-s − 1.35·39-s − 7.79·41-s − 11.1·43-s + 3.01·47-s + 9.42·49-s − 2.62·51-s − 5.03·53-s − 2.39·57-s + 10.9·59-s + 12.7·61-s − 11.6·63-s − 4.70·67-s + 0.924·69-s + ⋯
L(s)  = 1  + 0.195·3-s + 1.53·7-s − 0.961·9-s + 0.301·11-s − 1.10·13-s − 1.87·17-s − 1.62·19-s + 0.299·21-s + 0.568·23-s − 0.384·27-s − 0.878·29-s + 0.0394·31-s + 0.0590·33-s − 0.218·37-s − 0.217·39-s − 1.21·41-s − 1.69·43-s + 0.439·47-s + 1.34·49-s − 0.367·51-s − 0.692·53-s − 0.317·57-s + 1.42·59-s + 1.62·61-s − 1.47·63-s − 0.574·67-s + 0.111·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(17.5670\)
Root analytic conductor: \(4.19131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 - T \)
good3 \( 1 - 0.339T + 3T^{2} \)
7 \( 1 - 4.05T + 7T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + 7.74T + 17T^{2} \)
19 \( 1 + 7.06T + 19T^{2} \)
23 \( 1 - 2.72T + 23T^{2} \)
29 \( 1 + 4.73T + 29T^{2} \)
31 \( 1 - 0.219T + 31T^{2} \)
37 \( 1 + 1.32T + 37T^{2} \)
41 \( 1 + 7.79T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 - 3.01T + 47T^{2} \)
53 \( 1 + 5.03T + 53T^{2} \)
59 \( 1 - 10.9T + 59T^{2} \)
61 \( 1 - 12.7T + 61T^{2} \)
67 \( 1 + 4.70T + 67T^{2} \)
71 \( 1 + 2.52T + 71T^{2} \)
73 \( 1 - 4.10T + 73T^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 + 3.63T + 83T^{2} \)
89 \( 1 - 9.88T + 89T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.557155909161096661731403885108, −8.163035621783852194683067225804, −7.09662539989330492498518437249, −6.44478529920959154475673841054, −5.23263767041411721412702085866, −4.76401531178458325042431270961, −3.84386620845906273388540555579, −2.42536656326177769020911188109, −1.89512849568786469430926259679, 0, 1.89512849568786469430926259679, 2.42536656326177769020911188109, 3.84386620845906273388540555579, 4.76401531178458325042431270961, 5.23263767041411721412702085866, 6.44478529920959154475673841054, 7.09662539989330492498518437249, 8.163035621783852194683067225804, 8.557155909161096661731403885108

Graph of the $Z$-function along the critical line